You have just completed your registration at OpenAire.
Before you can login to the site, you will need to activate your account.
An e-mail will be sent to you with the proper instructions.
Important!
Please note that this site is currently undergoing Beta testing.
Any new content you create is not guaranteed to be present to the final version
of the site upon release.
We establish new existence results for multiple positive solutions of fourth-order nonlinear equations which model deflections of an elastic beam. We consider the widely studied boundary conditions corresponding to clamped and hinged ends and many non-local boundary conditions, with a unified approach. Our method is to show that each boundary-value problem can be written as the same type of perturbed integral equation, in the space $C[0,1]$, involving a linear functional $\alpha[u]$ but, although we seek positive solutions, the functional is not assumed to be positive for all positive $u$. The results are new even for the classic boundary conditions of clamped or hinged ends when $\alpha[u]=0$, because we obtain sharp results for the existence of one positive solution; for multiple solutions we seek optimal values of some of the constants that occur in the theory, which allows us to impose weaker assumptions on the nonlinear term than in previous works. Our non-local boundary conditions contain multi-point problems as special cases and, for the first time in fourth-order problems, we allow coefficients of both signs.
J. Ehme, P. Eloe and J. Henderson. Upper and lower solutions method for fully nonlinear boundary-value problems. J. Diff. Eqns 180 (2002), 51-64.
D. Franco, D. O'Regan and J. Per´an. Fourth-order problems with nonlinear boundary conditions. J. Computat. Appl. Math. 174 (2005), 315-327.
Applic. Analysis 72 (1999), 439-448.
J. R. Graef and B. Yang. Existence and nonexistence of positive solutions of fourth-order nonlinear boundary-value problems. Applic. Analysis 74 (2000), 201-214.
J. R. Graef, Ch. Qian and B. Yang. A three point boundary-value problem for nonlinear fourth-order differential equations. J. Math. Analysis Applic. 287 (2003), 217-233.
D. Guo and V. Lakshmikantham. Nonlinear problems in abstract cones (Academic, 1988).
Z. Hao and L. Debnath. On eigenvalue intervals and eigenfunctions of fourth-order singular boundary-value problems. Appl. Math. Lett. 18 (2005), 543-553.
G. Infante and J. R. L. Webb. Nonlinear non-local boundary-value problems and perturbed Hammerstein integral equations. Proc. Edinb. Math. Soc. 49 (2006), 637-656.
J. W. Jerome. Linear self-adjoint multi-point boundary-value problems and related approximation schemes. Numer. Math. 15 (1970), 433-449.
A. J. Jerri. Introduction to integral equations with applications, 2nd edn (New York: Wiley Interscience, 1999).
P. Korman. Uniqueness and exact multiplicity of solutions for a class of fourth-order semilinear problems. Proc. R. Soc. Edinb. A 134 (2004), 179-190.
M. A. Krasnosel'ski˘ı and P. P. Zabre˘ıko. Geometrical methods of nonlinear analysis (Springer, 1984).
J. Lond. Math. Soc. 63 (2001), 690-704.
K. Q. Lan and J. R. L. Webb. Positive solutions of semilinear differential equations with singularities. J. Diff. Eqns 148 (1998), 407-421.
F. Li, Q. Zhang, and Z. Liang. Existence and multiplicity of solutions of a kind of fourthorder boundary-value problem. Nonlin. Analysis 62 (2005), 803-816.
Computat. 148 (2004), 407-420.
Y. Liu and W. Ge. Double positive solutions of fourth-order nonlinear boundary-value problems. Applic. Analysis 82 (2003), 369-380.
F. Minh´os, T. Gyulov and A. I. Santos. Existence and location result for a fourth-order boundary-value problem. Discrete Cont. Dynam. Syst. Suppl. (2005), 662-671.
R. D. Nussbaum. Periodic solutions of some nonlinear integral equations. Dynamical Systems, Proc. Int. Symposium, University of Florida, Gainesville, FL, 1976, pp. 221-249 (Academic, 1977).
D. Smets and J. B. van den Berg. Homoclinic solutions for Swift-Hohenberg and suspension bridge type equations. J. Diff. Eqns 184 (2002), 78-96.
W. Soedel. Vibrations of shells and plates (New York: Dekker, 1993).
S. P. Timoshenko. Theory of elastic stability (New York: McGraw-Hill, 1961).
G. Wang, M. Zhou and L. Sun. Fourth-order problems with fully nonlinear boundary conditions. J. Math. Analysis Applic. 325 (2007), 130-140.
Discrete Contin. Dynam. Syst. Ser. S 1 (2008), 177-186.
J. R. L. Webb and G. Infante. Positive solutions of nonlocal boundary value problems involving integral conditions. Nonlin. Diff. Eqns Applic. (doi: 10.1007/s00030-007-4067-7.) J. R. L. Webb and K. Q. Lan. Eigenvalue criteria for existence of multiple positive solutions of nonlinear boundary-value problems of local and non-local type. Topolog. Meth. Nonlin.
Analysis 27 (2006), 91-116.
Z. Wei and C. Pang. The method of lower and upper solutions for fourth-order singular m-point boundary-value problems. J. Math. Analysis Applic. 322 (2006), 675-692.
Appl. Math. Lett. 17 (2004), 237-243.
Q. Yao. On the positive solutions of a nonlinear fourth-order boundary-value problem with two parameters. Applic. Analysis 83 (2004), 97-107.